Question: Simplify; express your answer in exponential form. Assume $z\neq 0, p\neq 0$. $\dfrac{{(z^{2})^{-2}}}{{(z^{5}p)^{3}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${z^{2}}$ to the exponent ${-2}$ . Now ${2 \times -2 = -4}$ , so ${(z^{2})^{-2} = z^{-4}}$ In the denominator, we can use the distributive property of exponents. ${(z^{5}p)^{3} = (z^{5})^{3}(p)^{3}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(z^{2})^{-2}}}{{(z^{5}p)^{3}}} = \dfrac{{z^{-4}}}{{z^{15}p^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-4}}}{{z^{15}p^{3}}} = \dfrac{{z^{-4}}}{{z^{15}}} \cdot \dfrac{{1}}{{p^{3}}} = z^{{-4} - {15}} \cdot p^{- {3}} = z^{-19}p^{-3}$.